Rigid Body Simulation - Carnegie Mellon School of …baraff/sigcourse/slidesd.pdf · ·...

SD1SIGGRAPH ’97 COURSE NOTES PHYSICALLY BASED MODELING

Rigid Body Simulation

David BaraffRobotics Institute andSchool of Computer Science

Rigid Body SimulationRigid Body Simulation

David BaraffDavid BaraffRobotics Institute andRobotics Institute andSchool of Computer ScienceSchool of Computer Science

CarnegieMellonCarnegieMellon


Particle MotionParticle MotionParticle Motion

x(t)x(t)

v(t)v(t)


Particle StateParticle StateParticle State

x(t)x(t) v(t)v(t)

Y =Y =

6 7 4 4 8 4 4 6 7 4 4 8 4 4 6 7 4 4 8 4 4 6 7 4 4 8 4 4

Y =x(t)

v(t)

Y =x(t)

v(t)


Particle DynamicsParticle DynamicsParticle Dynamics

F(t)F(t)

v(t)v(t)

x(t)x(t)


State DerivativeState DerivativeState Derivative

v(t)v(t)6 7 4 4 8 4 4 6 7 4 4 8 4 4 6 7 4 4 8 4 4 6 7 4 4 8 4 4

ddt

Y =ddt

x(t)

v(t)

=v(t)

F(t) / m

ddt

Y =ddt

x(t)

v(t)

=v(t)

F(t) / m

F(t) / mF(t) / m

d

dtY =

d

dtY =


Multiple ParticlesMultiple ParticlesMultiple Particles


State DerivativeState DerivativeState Derivative

6n elements6n elements...... ......d

dtY =

d

dtY =

ddt

Y =ddt

x1(t)

v1(t)

M

xn (t)

vn (t)

=

v1(t)

F1(t) / m1

M

vn (t)

Fn (t) / mn

ddt

Y =ddt

x1(t)

v1(t)

M

xn (t)

vn (t)

=

v1(t)

F1(t) / m1

M

vn (t)

Fn (t) / mn


ODE solutionODE solutionODE solution

Y(t0) Y(t0)

Y(t0 + ∆t) Y(t0 + ∆t)

...... Y(t1) Y(t1)

d

dtY(t0 )

d

dtY(t0 )

d

dtY(t0 + ∆t)

d

dtY(t0 + ∆t)


ODE solverODE solver

Y(t0) Y(t0)

t0 t0

t1 t1

dydtdydt

lenlen

void dydt(double t, double y[], double ydot[])

void dydt(double t, double y[], double ydot[])

Y(t1) Y(t1)


dydtdydtdydt

Y(t) = Y(t) =

x1(t)

v1(t)

xn(t)

vn(t)

x1(t)

v1(t)

xn(t)

vn(t)

v1(t)

F1(t)/ m1

vn(t)

Fn(t)/mn

v1(t)

F1(t)/ m1

vn(t)

Fn(t)/mn

d

dtY(t) =

d

dtY(t) =


Rigid Body StateRigid Body StateRigid Body State

v(t)v(t)

x(t)x(t)

Y =

x(t)

?

v(t)

?

Y =

x(t)

?

v(t)

?


Rigid Body Equation of MotionRigid Body Equation of MotionRigid Body Equation of Motion

ddt

Y =ddt

x(t)

?

Mv(t)

?

=

v(t)

?

F(t)

?

ddt

Y =ddt

x(t)

?

Mv(t)

?

=

v(t)

?

F(t)

?


Net ForceNet ForceNet Force

v(t)v(t)

x(t)x(t)

f1 f1

f2 f2

f3 f3

F(t) = fi ∑ F(t) = fi ∑


OrientationOrientationOrientation

We represent orientation as a rotation matrix R(t). Points are transformed from body-space to world-space as:

We represent orientation as a rotation matrix R(t). Points are transformed from body-space to world-space as:

††

††He’s lying. Actually, we use quaternions.He’s lying. Actually, we use quaternions.

p(t) = R(t)p0 + x(t)p(t) = R(t)p0 + x(t)


00

p(t)p(t)

body spacebody space world spaceworld space

x(t)x(t)

p0p0


Angular VelocityAngular VelocityAngular Velocity

We represent angular velocity as a vector ω(t), which encodes both the axis of the spin and the speed of the spin.

We represent angular velocity as a vector ω(t), which encodes both the axis of the spin and the speed of the spin.


Angular Velocity DefinitionAngular Velocity DefinitionAngular Velocity Definition

ω(t)ω(t)

x(t)x(t)

How are R(t) andω(t) related?How are R(t) andω(t) related?


Angular VelocityAngular VelocityAngular Velocity

R(t) and ω(t) are related by R(t) and ω(t) are related by ••

ddt

R(t) =0 −ωz (t) ω y (t)

ωz (t) 0 −ω x (t)

−ωy (t) ω x (t) 0

R(t)ddt

R(t) =0 −ωz (t) ω y (t)

ωz (t) 0 −ω x (t)

−ωy (t) ω x (t) 0

R(t)

(ω(t)* is a shorthand for the above matrix) (ω(t)* is a shorthand for the above matrix)



Need to relate ω(t) and mass distribution to F(t).Need to relate ω(t) and mass distribution to F(t).••

ddt

Y =ddt

x(t)

R(t)

Mv(t)

< ω(t) >

=

v(t)

ω(t)* R(t)

F(t)

?

ddt

Y =ddt

x(t)

R(t)

Mv(t)

< ω(t) >

=

v(t)

ω(t)* R(t)

F(t)

?


Inertia TensorInertia TensorInertia Tensor

†Integrals are precomputed.†Integrals are precomputed.

I(t) =Ixx Ixy Ix z

Iyx Iyy Iy z

Iz x Iz y Iz z

I(t) =Ixx Ixy Ix z

Iyx Iyy Iy z

Iz x Iz y Iz z

Ixx = M (y2 + z2 )dVV∫Ixx = M (y2 + z2 )dVV∫

off-diagonal terms†off-diagonal terms†diagonal terms†diagonal terms†

Ixy = −M xydVV∫Ixy = −M xydVV∫


Net TorqueNet TorqueNet Torque

x(t)x(t)

f1 f1

f2 f2

f3 f3

p1 p1

p2 p2

p3 p3

τ (t) = (pi − x(t)) × fi∑

τ (t) = (pi − x(t)) × fi∑


ddt

Y =ddt

x(t)

R(t)

Mv(t)

I(t)ω(t)

=

v(t)

ω(t)* R(t)

F(t)

τ (t)

ddt

Y =ddt

x(t)

R(t)

Mv(t)

I(t)ω(t)

=

v(t)

ω(t)* R(t)

F(t)

τ (t)


P(t) – linear momentum

L(t) – angular momentum

P(t) – linear momentum

L(t) – angular momentum


What’s in the Course NotesWhat’s in the Course NotesWhat’s in the Course Notes

1. Implementation of 1. Implementation of dydt dydt for rigid bodiesfor rigid bodies

(bookkeeping, data structures, computations) (bookkeeping, data structures, computations)

2. Quaternions – derivations and code2. Quaternions – derivations and code

3. Miscellaneous formulas and examples3. Miscellaneous formulas and examples

4. Derivations for force and torque equations,4. Derivations for force and torque equations,

center of mass, inertia tensor, rotation center of mass, inertia tensor, rotation

equations, velocity/acceleration of points equations, velocity/acceleration of points


ConstraintsConstraintsConstraints

We want rigid bodies to behave as solid objects, We want rigid bodies to behave as solid objects, and not inter-penetrate. By applying and not inter-penetrate. By applying constraintconstraint forces between contacting bodies, we prevent forces between contacting bodies, we prevent interpenetration from occurring. We need to:interpenetration from occurring. We need to:

a) Detect interpenetration a) Detect interpenetration

b) Determine contact points b) Determine contact points

c) Compute constraint forces c) Compute constraint forces


Simulations with CollisionsSimulations with CollisionsSimulations with Collisions

Y(t0) Y(t0)



Y(t0) Y(t0)

Y(t0 + ∆t) Y(t0 + ∆t)



Y(t0) Y(t0)

Y(t0 + ∆t) Y(t0 + ∆t)

Y(t0 +2∆t) Y(t0 +2∆t)


An Illegal State YAn Illegal State YAn Illegal State Y

Y(t0) Y(t0)

Y(t0 + ∆t) Y(t0 + ∆t)

Y(t0 +2∆t) Y(t0 +2∆t)

Y(t0 +3∆t) Y(t0 +3∆t)

illegalstate

illegalstate


Backing up to the Collision TimeBacking up to the Collision TimeBacking up to the Collision Time

Y(tc) Y(tc)


Colliding ContactColliding ContactColliding Contact

pa pa

AA

BB

n • pa < 0 n • pa < 0

pa pa n n


Resting ContactResting ContactResting Contact

pa pa

AA

BB

n n

pa pa

n • pa = 0 n • pa = 0


Collision ProcessCollision ProcessCollision Process

noforceno

forceno

forceno

force

∆t ∆t


A Soft CollisionA Soft CollisionA Soft Collision

forceforce velocityvelocity

∆t ∆t


A Harder CollisionA Harder CollisionA Harder Collision


∆t ∆t


A Very Hard CollisionA Very Hard CollisionA Very Hard Collision


∆t ∆t


A Rigid Body CollisionA Rigid Body CollisionA Rigid Body Collision

impulsive force

impulsive force

velocityvelocity

fimp = ∞

∆t = 0

fimp = ∞

∆t = 0


Colliding ContactColliding ContactColliding Contact

pa pa

AA

BB

pa pa n n

impulse(alters velocity)

impulse(alters velocity)


Resting ContactResting ContactResting Contact

pa pa

AA

BB

n n

pa pa

force(alters acceleration)

force(alters acceleration)

externalforces

externalforces


dydt for Solid Objectsdydtdydt for Solid Objects for Solid Objects

Updatecurrentstate

Updatecurrentstate

CollisiondetectionCollisiondetection

Contact pointdeterminationContact pointdetermination

Collision response Collision response

Constraint/friction forcedetermination

Constraint/friction forcedetermination

Penetrationdetected

Estimate:

Penetrationdetected

Estimate: DiscontinuityDiscontinuity

ExceptionsExceptions

Y(t), tY(t), t

tc tc

d

dtY(t)

d

dtY(t)


In the Course Notes – Collision DetectionIn the Course Notes – Collision DetectionIn the Course Notes – Collision Detection

Bounding box check between Bounding box check between nn objects: yes, you objects: yes, you cancan avoid avoid OO((nn22)) work. Don’t even settle for work. Don’t even settle for OO((nn log log n n) ) – insist on an – insist on an OO((nn) ) algorithm!algorithm!

A coherence based collision detection strategy A coherence based collision detection strategy for convex polyhedra: it’s simple, efficient and for convex polyhedra: it’s simple, efficient and (relatively) easy to program.(relatively) easy to program.


Computing ImpulsesComputing ImpulsesComputing Impulses

AA

BB

n n

jn jn

pa– pa–

pa+ = ? pa+ = ?


Coefficient of RestitutionCoefficient of RestitutionCoefficient of Restitution

AA

BB

n n

jn jn

pa– pa–

n • pa+

= – ( )ε n • pa– n • pa

+ = – ( )ε n • pa

–

ε = 1 ε = 1

ε = 12 ε = 12

ε = 0 ε = 0


Computing jComputing Computing jj

pa pa

AA

BB

n n

jn jn

cj + b = d cj + b = d n • pa+

= – ( )ε n • pa– n • pa

+ = – ( )ε n • pa

–



pa pa

AA

n n

BB pb pb jn jn

n • (pa+

– pb+

) = – ( )ε n • (pa– – pb

–) n • (pa+

– pb+

) = – ( )ε n • (pa– – pb

–)



pa pa

AA

n n

BB pb pb jn jn

n • (pa+

– pb+

) = – ( )ε n • (pa– – pb

–) n • (pa+

– pb+

) = – ( )ε n • (pa– – pb

–) cj + b = d cj + b = d


In the Course Notes – Collision ResponseIn the Course Notes – Collision ResponseIn the Course Notes – Collision Response

Data structures to represent contacts (found by Data structures to represent contacts (found by the collision detection phase).the collision detection phase).

Derivations and code for computing the impulse Derivations and code for computing the impulse between two colliding frictionless bodies for a between two colliding frictionless bodies for a particular coefficient of particular coefficient of ε.ε.

Code to detect collisions and apply impulses.Code to detect collisions and apply impulses.


Resting Contact ForcesResting Contact ForcesResting Contact Forces

pa pa

AA

BB

n n

externalforces

externalforces

f n f n


Conditions on the Constraint ForceConditions on the Constraint ForceConditions on the Constraint Force

To avoid inter-penetration, the force strength f must prevent the vertex pa from accelerating downwards. If B is fixed, this is written as

To avoid inter-penetration, the force strength f must prevent the vertex pa from accelerating downwards. If B is fixed, this is written as

n • pa ≥ 0 n • pa ≥ 0


Computing fComputing Computing ff

pa pa

AA

BB

n n

n • pa ≥ 0 n • pa ≥ 0

f n f n

a f + b ≥ 0 a f + b ≥ 0



To prevent the constraint force from holding bodies together, the force must be repulsive:

Does the above, along with

sufficiently constrain f ?

To prevent the constraint force from holding bodies together, the force must be repulsive:

Does the above, along with

sufficiently constrain f ?

f ≥ 0 f ≥ 0

n • pa ≥ 0 n • pa ≥ 0 a f + b ≥ 0 a f + b ≥ 0


Workless Constraint ForceWorkless Constraint ForceWorkless Constraint Force

pa pa

AA

BB

n n

Either

or

Either

or

f n f n

a f + b = 0 a f + b = 0

a f + b > 0 a f + b > 0

f ≥ 0

f ≥ 0

f = 0f = 0



To make f be workless, we use the condition

The full set of conditions is

To make f be workless, we use the condition

The full set of conditions is

f ⋅ (af + b) = 0 f ⋅ (af + b) = 0

f ⋅ (af + b) = 0 f ⋅ (af + b) = 0

f ≥ 0

f ≥ 0

a f + b ≥ 0 a f + b ≥ 0


Multiple Contact PointsMultiple Contact PointsMultiple Contact Points

AA

BB

CC

f1 n1 f1 n1

f2 n2 f2 n2


Conditions on f1Conditions on Conditions on f f11

Workless:Workless:

Non-penetration:Non-penetration: Repulsive:Repulsive:

a11 f1 + a12 f2 + b1 ≥ 0a11 f1 + a12 f2 + b1 ≥ 0 f1 ≥ 0 f1 ≥ 0

f1 ⋅ (a11 f1 + a12 f2 + b1) = 0 f1 ⋅ (a11 f1 + a12 f2 + b1) = 0


Workless:Workless:

Repulsive:Repulsive:

a11 f1 + a12 f2 + b1 ≥ 0

a21 f1 + a22 f2 + b2 ≥ 0

a11 f1 + a12 f2 + b1 ≥ 0

a21 f1 + a22 f2 + b2 ≥ 0 f1 ≥ 0

f2 ≥ 0

f1 ≥ 0

f2 ≥ 0

f1 ⋅ (a11 f1 + a12 f2 + b1) = 0

f2 ⋅ (a21 f1 + a22 f2 + b2) = 0

f1 ⋅ (a11 f1 + a12 f2 + b1) = 0

f2 ⋅ (a21 f1 + a22 f2 + b2) = 0

Non-penetration:Non-penetration:

Quadratic Program for f1 and f2Quadratic Program for Quadratic Program for f f11 and and ff22


In the Course Notes – Constraint ForcesIn the Course Notes – Constraint ForcesIn the Course Notes – Constraint Forces

Derivations of the non-penetration constraints for contacting polyhedra.

Derivations and code for computing the aij and bi coefficients.

Code for computing and applying the constraint forces .

Derivations of the non-penetration constraints for contacting polyhedra.

Derivations and code for computing the aij and bi coefficients.

Code for computing and applying the constraint forces . fi ni fi ni


Quadratic Programs with Equality ConstraintsQuadratic Programs with Equality ConstraintsQuadratic Programs with Equality Constraints

f1 ≥ 0

f2 ≥ 0

f1 ≥ 0

f2 ≥ 0

f1 ⋅ (a11 f1 + a12 f2 + b1) = 0

f2 ⋅ (a21 f1 + a22 f2 + b2) = 0

f1 ⋅ (a11 f1 + a12 f2 + b1) = 0

f2 ⋅ (a21 f1 + a22 f2 + b2) = 0

Non-penetration:Non-penetration: Repulsive:Repulsive:a11 f1 + a12 f2 + b1 0

a21 f1 + a22 f2 + b2 ≥ 0

a11 f1 + a12 f2 + b1 0

a21 f1 + a22 f2 + b2 ≥ 0

==

(free)(free)

Workless:Workless:

Rigid Body Simulation - Carnegie Mellon School of …baraff/sigcourse/slidesd.pdf · ·...

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